{"paper":{"title":"The Grothendieck group of non-commutative non-noetherian analogues of $\\mathbb{P}^1$ and regular algebras of global dimension two","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RA","authors_text":"Gautam Sisodia, S. Paul Smith","submitted_at":"2014-03-04T00:34:05Z","abstract_excerpt":"Let $V$ be a finite-dimensional positively-graded vector space. Let $b \\in V \\otimes V$ be a homogeneous element whose rank is $\\text{dim}(V)$. Let $A=TV/(b)$, the quotient of the tensor algebra $TV$ modulo the 2-sided ideal generated by $b$. Let ${\\sf gr}(A)$ be the category of finitely presented graded left $A$-modules and ${\\sf fdim}(A)$ its full subcategory of finite dimensional modules. Let ${\\sf qgr}(A)$ be the quotient category ${\\sf gr}(A)/{\\sf fdim}(A)$. We compute the Grothendieck group $K_0({\\sf qgr}(A))$. In particular, if the reciprocal of the Hilbert series of $A$, which is a pol"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1403.0640","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}